Brard 
1 
this notation recalls that any point M' on »s belongs to an arc 
BoB) 4. the abscissa X of By, being at t afunctionof M. 
‘ Let us consider a point My, Mz, or Mj on jo een or 
as . For instance M, determines a pair(X, t') and it belongs only to 
fe) 
the vortex disse pte (7) for 7<t' (Mg,) . The contribution from 
/ Sage is thus : 
e) 1 
(pi=T) =< —— dS(M, ). (5.45) 
a pe: X(M. ), t'(M MS la tice ‘ 
) 
f., f f fa E70 
! 
The contributions from vie and ae have analogous expressions. 
Lastly the contribution from the fifth part of Sx t! is given by 
t 
) 5 1 
Vy | Reesrar igen 9 pg 45 (M") (5.46) 
J. ff. Piss sek hh NiO Eauie Sree 
Remark - 
- The considerations of the present Subsection involve the im- 
plicit assumption that the positions of the lines , We 9 on the hull 
are independent of time. This may be untrue if the amplitude of the 
unsteady motions is great, and also if a strong separation occurs. 
- It is also to be pointed out that several free vortex sheets 
can exist without a rotational wake W,. This is, for instance, the 
case of biplanes. Each of the two wings gives rise to one vortex sheet 
and one deals with horseshoe free vortex filaments on each vortex 
sheet. 
VI - EQUATIONS DETERMINING THE TWO VORTEX FAMILIES (1) 
The fluid motion about the given body depends on the hull geo- 
metry, on the own motion of the body and on the incident flow on the 
body. The hull geometry is referred to a system S of axes moving 
with the body. The fluid motion is referred to a system S' 
of axes 
fixed in space. 
(1) See foot note on the following page. 
1230 
